A007497 -id:A007497 - cp's OEIS frontend

A175877 Positions of odd terms in A007497.

2, 4, 6, 33

Offset: 1

Zak Seidov, Oct 07 2010

Corresponding odd terms 3,7,15,564210119465811 (A175878).

n=2;Do[If[OddQ[n=DivisorSigma[1,n]],Print[{k,n}]],{k,2,99}]

A175878 Odd terms in A007497.

Original entry on oeis.org

3, 7, 15, 564210119465811

Offset: 1

Zak Seidov, Oct 07 2010

A007497, A175877.

n=2;Do[If[OddQ[n=DivisorSigma[1,n]],Print[n]],{k,2,99}]

A376422 Numbers m with largest nondivisor k <= m such that rad(k) | m is not powerful, where rad = A007497.

Original entry on oeis.org

24, 50, 54, 60, 75, 100, 102, 108, 112, 126, 165, 168, 170, 174, 180, 186, 189, 190, 192, 198, 200, 204, 216, 225, 231, 238, 242, 245, 315, 340, 363, 370, 374, 390, 396, 400, 402, 405, 408, 414, 416, 420, 426, 429, 432, 435, 442, 462, 465, 476, 480, 484, 490, 492

Offset: 1

Michael De Vlieger, Sep 22 2024

nonn

The term powerful used here refers to k in A001694, and rad = A007947.
Includes m such that the largest k = A373736(m) in row m of A272618 is not in A001694.
Subset of A024619, since for prime powers m = p^e, e >= 1, all k <= m such that rad(k) | m also divide m.
Subset of A376421, since nondivisor k such that rad(k) | m must be composite, and composite prime powers m in A246547 are a subset of A001694.

			6 is not included since nondivisor 4 = 2^2 is such that rad(4) | 6, but 4 is powerful since it is a perfect power of a prime.
24 is included since nondivisor 18 = 2 * 3^2 is such that rad(18) | 24 and is not powerful.
42 is not included since nondivisor 36 = 2^2 * 3^2 is such that rad(36) | 42 but 36 is powerful, since all exponents of prime power factors of 36 exceed 1, i.e., 36 is in A286708, a subset of A001694.
60 is in the sequence because nondivisor 54 = 2 * 3^3 but rad(54) | 60 and 54 is not powerful, etc.

Michael De Vlieger, Table of n, a(n) for n = 1..16464

Cf. A001694, A007947, A024619, A246547, A272618, A286708, A373736, A376421.

rad[x_] := Times @@ FactorInteger[x][[All, 1]];
Table[If[PrimePowerQ[n], Nothing,
  If[! Divisible[#, rad[#]^2], n, Nothing] &@
   SelectFirst[Range[n - 1, 1, -1],
    And[! Divisible[n, #], Divisible[n, rad[#]]] &] ], {n, 2, 500}]

A253274 Exponent of the highest power of 2 dividing A007497(n).

Original entry on oeis.org

1, 0, 2, 0, 3, 0, 3, 2, 3, 5, 3, 6, 2, 12, 3, 13, 3, 9, 6, 13, 10, 9, 7, 13, 9, 13, 5, 14, 6, 12, 11, 26, 0, 14, 15, 17, 6, 20, 15, 18, 17, 35, 10, 8, 23, 11, 15, 10, 16, 29, 32, 12, 19, 27, 25, 26, 18, 24, 33, 22, 36, 13, 24, 18, 24, 43, 16, 29, 35, 18, 12, 42

Offset: 1

Allan C. Wechsler, May 01 2015

nonn

			A007497(9) = 168 = 2^3*3*7, so a(n) = 3.

Amiram Eldar, Table of n, a(n) for n = 1..1500

Cf. A007497, A007814.

f[1] = 2; f[n_] := f[n] = DivisorSigma[1, f[n - 1]]; Table[IntegerExponent[f[n], 2], {n, 70}] (* Amiram Eldar, Mar 27 2023 *)

a(n) = A007814(A007497(n)).

More terms from Michel Marcus, May 01 2015

A333986 Greatest prime factor of A007497(n).

Original entry on oeis.org

2, 3, 2, 7, 2, 5, 3, 5, 7, 5, 7, 5, 127, 7, 8191, 5, 127, 13, 31, 127, 127, 89, 31, 17, 127, 31, 3221, 179, 151, 127, 8191, 13, 262657, 11939, 199, 257, 127, 127, 337, 257, 524287, 73, 1093, 547, 137, 241, 1093, 547, 331, 131071, 1093, 599479, 8191, 137, 127, 8191

Offset: 1

Richard Peterson, Sep 04 2020

nonn

Amiram Eldar, Table of n, a(n) for n = 1..1000

Cf. A006530, A007497.

f[1] = 2; f[n_] := f[n] = DivisorSigma[1, f[n - 1]]; Table[FactorInteger[f[n]][[-1, 1]], {n, 50}] (* Amiram Eldar, Sep 23 2020 *)

a(n) = A006530(A007497(n)). - Michel Marcus, Sep 09 2020

More terms from Michel Marcus, Sep 16 2020

A051572 a(1) = 5, a(n) = sigma(a(n-1)).

Original entry on oeis.org

5, 6, 12, 28, 56, 120, 360, 1170, 3276, 10192, 24738, 61440, 196584, 491520, 1572840, 5433480, 20180160, 94859856, 355532800, 1040179456, 2143289344, 4966055344, 10092086208, 31800637440, 137371852800, 641012414823

Offset: 1

Judson Neer

N. J. A. Sloane, Table of n, a(n) for n = 1..523
G. L. Cohen and H. J. J. te Riele, Iterating the sum-of-divisors function, Experimental Mathematics, 5 (1996), pp. 91-100.
Graeme L. Cohen and Herman J. J. te Riele, Iterating the Sum-of-Divisors Function, Experimental Mathematics, Vol. 5 (1996), No. 2, pp. 91-100.

Cf. A000203, A007497, A257348.

NestList[Plus@@Divisors[#] &, 5, 25] (* Alonso del Arte, Apr 28 2011 *)

a=[5];for(i=2,10,a=concat(a,sigma(a[#a]))); a \\ Charles R Greathouse IV, Oct 03 2011

from itertools import accumulate, repeat # requires Python 3.2 or higher
from sympy import divisor_sigma
A051572_list = list(accumulate(repeat(5,100), lambda x, _: divisor_sigma(x)))
# Chai Wah Wu, May 02 2015

A129246 Iterated sum of divisors array A[k,n] = k-th iterate of sigma(n), by upward antidiagonals.

Original entry on oeis.org

1, 1, 3, 1, 4, 4, 1, 7, 7, 7, 1, 8, 8, 8, 6, 1, 15, 15, 15, 12, 12, 1, 24, 24, 24, 28, 28, 8, 1, 60, 60, 60, 56, 56, 15, 15, 1, 168, 168, 168, 120, 120, 24, 24, 13, 1, 480, 480, 480, 360, 360, 60, 60, 14, 18, 1, 1512, 1512, 1512, 1170, 1170, 168, 168, 24, 39, 12, 1, 4800, 4800

Offset: 1

Jonathan Vos Post, May 27 2007

			Array begins:
k / sigma(...sigma(n)..) nested k deep.
1.|.1...3...4....7....6....12....8....15...13....18...
2.|.1...4...7....8...12....28...15....24...14....39...
3.|.1...7...8...15...28....56...24....60...24....56...
4.|.1...8..15...24...56...120...60...168...60...120...
5.|.1..15..24...60..120...360..168...480..168...360...
6.|.1..24..60..168..360..1170..480..1512..480..1170...
7.|.1..60.168..480.1170..3276.1512..4800.1512..3276...
8.|.1.168.480.1512.3276.10192.4800.15748.4800.10192...

Robert Israel, Table of n, a(n) for n = 1..10011 (first 142 antidiagonals, flattened)

Cf. A000203 (row 1), A051027 (row 2), A066971 (row 3).

Cf. A000012 (column 1), A007497 (column 2), A090896 (main diagonal).

A129246 := proc(k,n) option remember ; if k= 1 then numtheory[sigma](n); else A129246(k-1,numtheory[sigma](n)) ; fi ; end: for d from 1 to 13 do for n from 1 to d do printf("%d, ",A129246(d+1-n,n)) ; od: od: # R. J. Mathar, Oct 09 2007

T[n_, k_] := T[n, k] = If[n == 1, DivisorSigma[1, k], DivisorSigma[1, T[n-1, k]]];
Table[T[d-k+1, k], {d, 1, 13}, {k, 1, d}] // Flatten (* Jean-François Alcover, Sep 23 2022, after R. J. Mathar, except that T(n,k) replaces the unusual A(k,n) *)

A[k,n] = sigma^k(n), where sigma^k denotes functional iteration.

More terms from R. J. Mathar, Oct 09 2007

A257348 Repeatedly applying the map x -> sigma(x) partitions the natural numbers into a number of disjoint trees; sequence gives the (conjectural) list of minimal representatives of these trees.

Original entry on oeis.org

1, 2, 5, 16, 19, 27, 29, 33, 49, 50, 52, 66, 81, 85, 105, 146, 147, 163, 170, 189, 197, 199, 218, 226, 243, 262, 303, 315, 343, 424, 430, 438, 453, 461, 463, 469, 472, 484, 489, 513, 530, 550, 584, 677, 722, 746, 786, 787, 804, 813, 821, 831, 842, 859, 867, 876, 892, 903, 914, 916, 937, 977, 982, 988, 990, 1029

Offset: 1

N. J. A. Sloane, May 01 2015, following a suggestion from Kerry Mitchell

Very little is known for certain. Even the trajectories of 2 (A007497) and 5 (A051572) under repeated application of the map x -> sigma(x) (cf. A000203) are only conjectured to be disjoint.
The thousand-term b-file (up to 141441) has been checked to correspond to disjoint trees for 265 iterations of sigma on each term, and every non-term n < 141441 merges (in at most 21 iterations) with an earlier iteration sequence. - Hans Havermann, Nov 22 2019
Rather than trees we mean connected components of the graphs with edges x -> sigma(x). The number 1 is a fixed point, i.e., a cycle of length 1 under iterations of sigma, it is not part of a tree. But since sigma(n) > n for n > 1 there are no other cycles. - M. F. Hasler, Nov 21 2019

Kerry Mitchell, Posting to Math Fun Mailing List, Apr 30 2015

Hans Havermann, Table of n, a(n) for n = 1..1000
G. L. Cohen and H. J. J. te Riele, Iterating the sum-of-divisors function, Experimental Mathematics, 5 (1996), pp. 91-100. See Eq. (4.2).

Cf. A000203 (sigma), A007497 (trajectory of 2), A051572 (trajectory of 5), A257349 (trajectory of 16).

Cf. A216200 (number of disjoint trees up to n); A257669 and A257670: size and smallest number of subtree rooted in n.

More terms from Hans Havermann, May 02 2015

A257670 Minimum term in the sigma(x) -> x subtree whose root is n.

Original entry on oeis.org

1, 2, 2, 2, 5, 5, 2, 2, 9, 10, 11, 5, 9, 9, 2, 16, 17, 10, 19, 19, 21, 22, 23, 2, 25, 26, 27, 5, 29, 29, 16, 16, 33, 34, 35, 22, 37, 37, 10, 27, 41, 19, 43, 43, 45, 46, 47, 33, 49, 50, 51, 52, 53, 34, 55, 5, 49, 58, 59, 2, 61, 61, 16, 64, 65, 66, 67, 67, 69

Offset: 1

Michel Marcus, May 03 2015

nonn

			We have the following trees (a <- b means sigma(a) = b):
  2 <-- 3 <-- 4 <-- 7 <-- 8 <-- 15 <-- 24 <-- 60 <-- ...
                    9 <-- 13 <-- 14 <-’
  5 <-- 6 <-- 12 <-- 28 <-- 56 <-- 120 <-- ...
        11 <-’             /
       10 <-- 18 <-- 39 <-’
The number 1 has strictly speaking an arrow to itself, so it is not part of a tree. (For all n > 1, sigma(n) > n, so no other fixed point or longer "cycle" can exist.) But actually we rather consider connected components, and let a(1) = 1 as the smallest element of this connected component.
a(2) = 2, since there is no smaller x such that sigma(x) = 2: the subtree with root 2 is reduced to a single node: 2. Similarly, a(m) = m for all m in A007369.
For n=3, since sigma(2) = 3, the tree whose root is 3 has 2 nodes: 2 and 3, and the smallest one is 2, hence a(3) = 2.
Similarly, although 24 occurs directly first at sigma(14), it is also reached from 15 which is in turn reached, via intermediate steps, from 2. Thus, the subtree with root 24 has as 2 as smallest element, whence a(24) = 2.

M. F. Hasler, Table of n, a(n) for n = 1..20000, Nov 19 2019 (first 1000 terms from Michel Marcus)
M. Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems: invsigma.gp, Oct. 2005
G. L. Cohen and H. J. J. te Riele, Iterating the sum-of-divisors function, Experimental Mathematics, 5 (1996), pp. 91-100.
R. J. Mathar, Illustrations

Cf. A000203 (sigma), A007369  (sigma(x) = n has no solution).
Cf. A216200 (number of disjoint trees), A257348 (minimal node of all trees).
Cf. A257669 (number of terms in current tree).

lista(nn) = {my(v = vector(nn)); v[1] = 1; for (i=2, nn, my(s = i); while (s <= nn, if (v[s] == 0, v[s] = i); s = sigma(s););); for (i=1, nn, if (v[i] == 0, v[i] = i);); v;} \\ Michel Marcus, Nov 19 2019

A257670(n)=if(n>2,vecmin(concat(apply(self,invsigma(n)),n)),n) \\ See Alekseyev-link for invsigma(). - David A. Corneth and M. F. Hasler, Nov 20 2019

a(m) = m for m in A007369: sigma(x) = m has no solution. [Corrected by M. F. Hasler, Nov 19 2019]

a(A007497(n)) = 2; a(A051572(n)) = 5; a(A257349(n)) = 16. (These sequences being the trajectory of 2, 5 resp. 16 under iterations of sigma = A000203.)

Edited by M. F. Hasler, Nov 19 2019

A257349 a(1) = 16, a(n) = sigma(a(n-1)).

Original entry on oeis.org

16, 31, 32, 63, 104, 210, 576, 1651, 1792, 4088, 8880, 28272, 79360, 196416, 633984, 1827840, 7074432, 22032000, 86640840, 364989240, 1651141800, 7540142400, 33541980160, 90193969152, 334471118520, 1415960985600, 6118878991680, 29424972595200

Offset: 1

N. J. A. Sloane, May 01 2015

nonn

Chai Wah Wu, Table of n, a(n) for n = 1..1000

Cf. A000203, A007497, A051572, A257348.

NestList[DivisorSigma[1,#]&,16,27] (* Ivan N. Ianakiev, May 02 2015 *)

lista(nn) = {print1(v = 16, ", "); for (n=1, nn, v = sigma(v); print1(v, ", "););} \\ Michel Marcus, May 02 2015

from itertools import accumulate, repeat # requires Python 3.2 or higher
from sympy import divisor_sigma
A257349_list = list(accumulate(repeat(16,100), lambda x, _: divisor_sigma(x)))
# Chai Wah Wu, May 02 2015

cp's OEIS Frontend

A175877 Positions of odd terms in A007497.

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A175878 Odd terms in A007497.

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A376422 Numbers m with largest nondivisor k <= m such that rad(k) | m is not powerful, where rad = A007497.

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A253274 Exponent of the highest power of 2 dividing A007497(n).

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A333986 Greatest prime factor of A007497(n).

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A051572 a(1) = 5, a(n) = sigma(a(n-1)).

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A129246 Iterated sum of divisors array A[k,n] = k-th iterate of sigma(n), by upward antidiagonals.

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A257348 Repeatedly applying the map x -> sigma(x) partitions the natural numbers into a number of disjoint trees; sequence gives the (conjectural) list of minimal representatives of these trees.

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A257670 Minimum term in the sigma(x) -> x subtree whose root is n.

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